在本論文中，我們研讀對應高斯擴散過程的正值時間空間調合函數之積分表現式。高斯擴散過程$X_{t}$在$\mathbb{R}^{d}$上面滿足
\[
\begin{cases}
dX_{t}=BX_{t}dt+dW_{t},\\
X_{0}=x_{0},
\end{cases}
\]
其中$B$是個$d\times d$矩陣， $W$是個$d$維布朗運動，而$x_{0}\in\mathbb{R}^{d}$是$X$的初始值。$g$是個正值時間空間調合函數對應到隨機過程$X_{t}$且滿足
\begin{align*}
(\frac{\partial}{\partial t}+\frac{1}{2}\triangle+Bx\cdot\nabla)g=0,\mbox{ }g>0\mbox{ on }(0,\infty)\times\mathbb{R}^{d}.
\end{align*}
$g$的積分表現式是
\begin{align*}
g(t,x) & =\int_{\mathbb{R}^{d}}M_{B}(t,x;z)\rho(dz),
\end{align*}
其中$\rho$是個機率分布且$\{M_{B}(\cdot,\cdot;z);z\in\mathbb{R}^{d}\}$是一系列獨立於$g$的函數。為了獲得表現式，我們考慮一個跟$g$有關的隨機過程$X_{t}^{g}$，其中$X_{t}^{g}$滿足
\[
\begin{cases}
dX_{t}^{g}=\frac{\nabla g(t,X_{t}^{g})}{g(t,X_{t}^{g})}dt+B\cdot X_{t}^{g}dt+dW_{t},\\
X_{0}^{g}=x_{0}.
\end{cases}
\]
我們研究$X_{t}^{g}$的極限行為當$t\rightarrow\infty$。我們先得到一個有趣的$X_{t}^{g}$表現式。然後$g$的積分表現式自然可以從$X_{t}^{g}$表現式獲得。在第一部分，我們考慮布朗運動，也就是$B=0$。在這個案例裡，我們證明$X_{t}^{g}$有線性成長速度$Y$當$t\rightarrow\infty$。也就是說
\begin{align*}
\frac{X_{t}^{g}}{t}\rightarrow Y,\mbox{ as }t\rightarrow\infty,
\end{align*}
其中$Y$是個隨機變數。此外，$X_{t}^{g}$有令人意外的表現式
\begin{align*}
X_{t}^{g} & =x_{0}+tY+\widehat{W}_{t},
\end{align*}
其中$\widehat{W}_{t}$是個獨立於$Y$的布朗運動。利用這個結果，我們獲得$g$的積分表現式，其中$\rho$(在表現式裡)是$Y$的機率分布。在第二部分，我們考慮一般的$B$。我們利用類似的方法去獲得$X_{t}^{g}$的不同成長速度和$X_{t}^{g}$表現式。然後我們可以得到$g$的積分表現式。我們也討論一些時間空間調合函數的積分表現式的應用。第一個例子是看正值(空間)調合函數的積分表現式。第二個例子是用來看邊界穿越機率的計算。

In this dissertation, we study the integral representation of positive
space-time harmonic function for Gaussian diffusion processes. A Gaussian
diffusion process $Y_{t}$ in $\mathbb{R}^{d}$ is governed by
\[
\begin{cases}
dY_{t}=BY_{t}dt+dW_{t},\\
Y_{0}=x_{0},
\end{cases}
\]
where $B$ is a $d\times d$ matrix, $W$ is a $d-$dimensional Brownian
motion, and $x_{0}\in\mathbb{R}^{d}$ is the initial value of $Y$.
$g$ is a positive space-time harmonic function for $Y_{t}$ which
satisfies
\begin{align*}
(\frac{\partial}{\partial t}+\frac{1}{2}\triangle+Bx\cdot\nabla)g=0,\mbox{ }g>0\mbox{ on }(0,\infty)\times\mathbb{R}^{d}.
\end{align*}
The integral formula of $g$ is given by
\begin{align*}
g(t,x) & =\int_{\mathbb{R}^{d}}M_{B}(t,x;z)\rho(dz),
\end{align*}
where $\rho$ is a probability distribution and $\{M_{B}(\cdot,\cdot;z);z\in\mathbb{R}^{d}\}$
is a family of functions which is independent of $g$. To obtain such
integral representation, we consider a process associated to $g$
deduced by $X_{t}$ which is governed by
\[
\begin{cases}
dX_{t}=\frac{\nabla g(t,X_{t})}{g(t,X_{t})}dt+B\cdot X_{t}dt+dW_{t},\\
X_{0}=x_{0}.
\end{cases}
\]
We study the limiting behavior of $X_{t}$ as $t\rightarrow\infty$.
We first obtain an interesting representation of $X_{t}$. Then the
integral formula of $g$ will follow. In Part 1, we consider the Brownian
motion, where $B=0$. In this case, we show $X_{t}$ has linear growth
with the rate given by $Y$ as $t\rightarrow\infty$. This means
\begin{align*}
\frac{X_{t}^{g}}{t}\rightarrow Y,\mbox{ as }t\rightarrow\infty,
\end{align*}
where $Y$ is a random variable. Futhermore, $X_{t}$ has remarkable
representation
\begin{align*}
X_{t} & =x_{0}+tY+\widehat{W}_{t},
\end{align*}
where $\widehat{W}_{t}$ is a Brownian motion independent of $Y$.
Using this, we obtain an integral representation for $g$, where $\rho$
(in the representation) is the disrtibution of $Y$. In Part 2, we
consider general $B$. We apply the similar approach to obtain the
growth of $X_{t}$, with different rate and a representation of $X_{t}$.
Then we can obtain the integral representation formula of $g$. We
also discuss some applications of the integral representation of space-time
harmonic functions. The first example is the integral representation
for a positive (space) harmonic functions. The second example is the
use in the calculation of the boundary crossing probability.

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